Numerical study of a hybrid membrane cell with semi and fully permeable membrane sub-sections

Chemical Engineering Science 62 (2007) 1215 – 1229 www.elsevier.com/locate/ces

Numerical study of a hybrid membrane cell with semi and fully permeable membrane sub-sections J.M. Miranda a,∗ , J.B.L.M. Campos b a Centro de Engenharia Biológica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal b Centro de Estudos de Fenómenos de Transporte, Departamento de Eng. Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto

Frias, 4200-465 Porto Portugal Received 24 January 2006; received in revised form 31 August 2006; accepted 1 November 2006 Available online 10 November 2006

Abstract Hybrid membrane cells with up to 128 sections, each one comprising a fully and a semi-permeable membrane sub-section and, the limit case of a cell with an infinite number of membrane sections were studied by numerical methods. These hybrid cells separate a feed stream into two parts: a solvent stream which crosses the semi-permeable membranes and a concentrate stream which crosses the fully permeable membranes. The concentrate stream has a cleaning effect on the mass boundary layer over the semi-permeable membranes. The numerical results show that concentration polarization in hybrid cells is much lower than the polarization in conventional cells. Additionally, a highly concentrated solution is recovered. The cell with an infinite number of membrane sections (n) has the best performance: the lowest polarization and the highest concentration in the concentrate stream. As n increases to infinite, the concentration in the concentrate stream tends to the concentration over the semi-permeable membrane, i.e., to the maximum concentration inside the mass boundary layer. The number of membrane sections needed to achieve a performance similar to that of a cell with an infinite number of sections is very high, greater than 128. The velocity of the concentrate stream also plays an important role. As this velocity is increased (until an upper limit), the cleaning effect of the boundary layer intensifies but the purity of the concentrate stream decreases (dilution effect). An intermediate value for the velocity of the concentrate stream (between the lower and upper limit) should be used to optimize both effects. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Membranes; Laminar flow; Mass transfer; Simulation; Computational fluid dynamics; Separations

1. Introduction Cross-flow pressure-driven membrane processes such as reverse osmosis and ultrafiltration are useful to purify solvents, to concentrate solutions and to fractionate components. However, their efficiency is limited by the increase of solute concentration along the bulk of the retentate and by concentration polarization in the vicinity of the membrane. Solute concentration increases along the bulk because the solute is rejected by the membrane. The osmotic pressure in the bulk increases, and therefore the permeate velocity and the efficiency of the

∗ Corresponding author. Tel.: +351966540930.

E-mail addresses: [email protected] (J.M. Miranda), [email protected] (J.B.L.M. Campos). 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.11.001

separation process decrease. Concentration polarization develops when the solute rejected by the membrane diffuses slowly away from the membrane and accumulates, mainly in the vicinity of the membrane surface, increasing the thickness of the mass boundary layer along the tangential direction. As the concentration at the membrane surface increases, the osmotic pressure increases and the permeate velocity, Sherwood number and membrane selectivity decrease. Selectivity is an important problem in protein fractionation (van Reis et al., 1997; Cheang and Zydney, 2004; Ghosh, 2003). In this process, one of the proteins preferentially permeates the membrane, while the other is partially rejected. During polarization, the concentration of the rejected protein increases in the vicinity of the membrane and, because the permeation of this rejected protein increases with its concentration, the selectivity of the separation process decreases.

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Fig. 1. Hybrid membrane system to recover phenol from wastewaters (Ray et al., 1991) comprising a pervaporation unit operation which recovers the solute (phenol) and a reverse osmosis unit operation which recovers the solvent (water).

Feed stream

Reverse osmosis membrane

Pervaporation membrane

Fig. 2. Example of a hybrid membrane cell to separate volatile organic compounds (VOCs) from water. The cell comprises pervaporation membranes permeable to VOCs and reverse osmosis membranes permeable to water.

Since concentration polarization has so many undesirable effects, several mechanisms have been proposed to minimize it: turbulence promoters (Poyen et al., 1987), impinging jets (Miranda and Campos, 2000), backflushing (Kuberkar and Davis, 2001; Levesley and Hoare, 1999), Dean vortices (Chung et al., 1993) and pulse flow (Kennedy et al., 1974). Ironically, concentration polarization is a state of high solute/solvent separation. The solute concentration in the boundary layer is much higher than the solute concentration in the bulk of the retentate, and so an effective separation is achieved. However, some of the proposed mechanisms have the unintended consequence of mixing otherwise separated components. Some authors have proposed solutions that take advantage of this effective state of separation. These solutions are based in the use of hybrid membrane systems (Fig. 1) or hybrid membrane cells (Fig. 2). Hybrid membrane systems combine two membrane unit operations. Usually, one unit operation performs a solvent removal operation and the other a solute removal operation. Hybrid membrane cells combine membranes with different properties, frequently alternating in series. Binning (1961) was the first, as far as we know, to propose using a cell with two different membranes, each of which was permeable to one of the components of a binary mixture. In the cell proposed by Binning, the retentate concentration remains equal to the feed concentration. Shaw et al. (1972) analytically solved the flow and mass transport equations and concluded that in reverse osmosis productivity can increase by the use

of intermediate non-rejecting membrane sections. Lee and Lightfoot (1974), Schubert and Todd (1980) and Mitrovic and Radovanovic (1984) proposed similar ideas. Lee and Lightfoot made some preliminary numerical simulations of an ultrafiltration cell with removal of fluid from the boundary layer. They found that the efficiency of this hybrid cell is higher than the efficiency of an analogous conventional cell. Schubert and Todd proposed the recovery of solute-rich fluid from the boundary layer at several locations along the membrane. Using numerical methods, Mitrovic and Radovanovic studied a cell with two different kinds of membranes alternating in series. Ray et al. (1991) studied hybrid membrane systems and showed that the use of complementary unit operations (solute removal and solvent removal unit operations) in the same system can improve separation efficiency because each unit can operate at optimized conditions. Nitsche and Zhuge (1995) studied anti-polarization dialysis, a process based in a hybrid cell with two membranes, a longitudinal dialysis membrane permeable to the solute and a transversal membrane impermeable to the solute but permeable to the solvent. Ideally, the polarization of the transversal membrane and the polarization of the longitudinal membrane should cancel each other (Nitsche and Zhuge, 1995). This paper presents a numerical study of the separation of a solute from its solvent, in a hybrid cell with semi and fully permeable membrane sub-sections alternating in series. The process takes advantage of the highly concentrated fluid in the polarized boundary layer to achieve a better separation

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

and to reduce polarization effects on membrane efficiency. High solute recovery and low polarization levels are achieved. Since we are mainly interested in the study of hybrid membrane cells, some physical phenomena were neglected. The properties of the fluid were considered to be concentration independent and interactions between solute and membrane were not taken into account. This hybrid membrane cell is more complex than conventional cells and more difficult to manufacture. Existing membranes could in principle be used to build the cell presented in this work by joining together membranes of different pore sizes. But one should expect that such approach would lead to leakage problems in the junctions between membranes. Specially designed membranes must be developed to avoid this problem. Recent advances in micro-fabrication (van Rijn, 2002) have opened up the possibility of developing new compact and complex cells and some recent developments have shown that it is possible to make membranes from carbon nanotubes (Hinds et al., 2004) and membranes with regions of different permeability (Liu et al., 2003).

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The cell is divided into sections: the inlet section, the outlet section and n membrane sections. The length of the inlet section is Lin , the total length of all membrane sections is Lm and the length of the outlet section is Lout . The inlet and the outlet sections are impermeable both to the solute and to the solvent. They were introduced so that the discontinuities in the permeate velocity and the tangential diffusion of solute in the edges of the membrane could be taken into account. Each membrane section is composed of a semi-permeable membrane sub-section and a fully permeable membrane sub-section. Semi-permeable and fully permeable membrane sub-sections are numbered from 1 to n. The length of a semi-permeable sub-section is Ls and the length of a fully permeable sub-section is Lf . The geometric variables were normalized by the distance between plates L Lm Lin Lout , lm = , lin = , lout = , H H H H Lf Ls ls = , lf = . (1) H H Since the purity of the concentrate stream, P, the solvent stream velocity, Vms , and the solute concentration at the surface s , may change along of the semi-permeable sub-sections, Cm the length of each membrane section and from membrane section to membrane section, it is necessary to define mean variables: l=

2. Cell description The cell studied is schematically represented in Fig. 3. It is a parallel plate cell with two kinds of membrane alternating along the walls: one semi-permeable (impermeable to the solute and permeable to the solvent) and another, fully permeable (permeable to both solute and solvent). The distance between the cell walls is H, the width of the cell is W and the length of the cell is L. The feed stream is separated into three streams, a retentate stream and two permeate streams. One of the permeate streams crosses the semi-permeable membranes, and since it is almost pure solvent, it is called the solvent stream. The other one crosses the fully permeable membranes, and being rich in solute, it is called the concentrate stream. The two permeate streams are collected in two independent permeate chambers; the solvent chamber and the concentrate chamber. The mean velocity of the feed is V0 and the concentration of solute in the feed is C0 .

• The mean purity of the concentrate stream for section i, P¯i :  Lf Cm (Xf , 0) VZ (Xf , 0) dXf ¯ Pi = 0 , (2)  Lf 0 Vz (Xf , 0) dXf where Xf is the longitudinal coordinate with origin at the beginning of the fully permeable sub-section i. • The mean purity of the concentrate stream for the entire cell, P¯ , is equal to the arithmetic mean of the mean purity of the concentrate for the sub-sections n P¯i ¯ P = i=1 . (3) n

Solvent stream Concentrate stream

Lin

Lm

Lout

Feed

Retentate z

x

Membrane section

Ls

Lf

Concentrate stream Solvent stream Fig. 3. Schematic representation of the hybrid cell studied.The cell has an inlet section, an outlet section and n membrane sections. Each membrane section has a semi-permeable sub-section and a fully permeable sub-section. In the permeate side, the cell has two independent collectors, one for the fluid that crosses the semi-permeable membranes and another to the fluid that crosses the fully permeable membranes.

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• The mean velocity of the solvent stream for sub-section i, s V mi , is given by  Ls

s

V mi =

0

VZ (Xs , 0) dXs ,  Ls 0 dXs

(4)

• The mean solute concentration at the surface of the semis permeable membrane of section i, C mi , is given by  Ls =

0

Cm (Xs , 0) dXs .  Ls dX s 0

(6)

• The mean solute concentration at the surface of all semipermeable membranes, C m , is equal to the arithmetic mean of the mean solute concentration for the sub-sections m s i=1 C mi Cm = . (7) m From here on, the velocities are adimensionalized by the mean feed velocity, V0 , and the concentrations by the feed concentration, C0 . Dimensionless variables are written in lowercase letters. 3. Numerical study The flow and the mass transport in the cell were studied by numerical methods. The equations, domain and respective boundary conditions were selected. The equations were discretized, and then solved to determine the flow and concentration fields. 3.1. Domain and numerical grid The number of grid nodes needed to accurately solve the flow and mass transport equations increases with the number of membrane sections. As the number of nodes of the grid increases, the computational resources required become too high. This is why the equations for cells with four or fewer membrane sections were solved simultaneously, and cells with more than four membrane sections were solved in a sequential section-by-section procedure. For cells with four or fewer membrane sections, domain A was used (Fig. 4). Domain A corresponds to half of the physical cell (the cell is symmetric) and has an inlet section, an outlet section and n membrane sections, each with two sub-sections. The fully permeable sub-sections are usually shorter than the

VII

Domain A

z

x III

VI IV

where Xs is the longitudinal coordinate with origin at the beginning of the semi-permeable sub-section i. • The mean velocity of the solvent stream for the entire cell, s V m , is equal to the arithmetic mean of the mean solvent velocity for the sub-sections: n s s i=1 V mi Vm = . (5) n

s C mi

I

V

Fig. 4. Schematic representation of domain A (cell with four or less membrane sections) and respective boundaries.

Table 1 Grid tests for n = 4 Grid

ni in

ni out

ni s

ni f

P

83 × 251 165 × 251 329 × 251 657 × 251 1313 × 251

3 3 5 9 17

3 3 5 9 17

13 17 49 97 193

9 25 33 65 129

21.6% 12.3% 6.1% 2.3% Ref

The variable ni in is the number of nodes in the longitudinal direction for the inlet section, ni out is the number of nodes in the longitudinal direction for the outlet section, ni s is the number of nodes in the longitudinal direction for each semi-permeable sub-section and ni f is the number of nodes in the longitudinal direction for each fully permeable sub-section. The results were obtained for the worst conditions studied (0 = 0.1; sv = 10−4 ; fv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

permeable ones. The domain was discretized and converted into a grid of discrete nodes. The mean purity of the concentrate is the result most sensitive to the grid density along the longitudinal direction. Table 1 shows the error of the purity of the concentrate for n = 4, for the worst conditions considered in this paper, and for several grid densities. The error was determined by   P − P    ref P =  (8)   P ref  in which P ref is the mean purity of the concentrate for a grid with 1297 × 251 nodes. The table shows that, a minimum of 65 nodes along the longitudinal direction is needed for each fully permeable sub-section to obtain an error less than 3%. A grid with 657 × 251 nodes was selected and is represented in Fig. 5a. Part of the grid was magnified and is represented in Fig. 5b. The grid has a variable density that increases wherever the concentration gradients are high: in the vicinity of the membrane and near the boundaries between sections. For cells with more than four membrane sections, the flow and mass transport equations were solved sequentially, one membrane section at a time. Three different domains (B, C and D), sketched in Fig. 6, were used. Domain B was used to solve the flow and mass transport equations for the first membrane section, domain C for intermediate membrane sections and domain D for the last membrane section. Consecutive domains have common regions so that mass transport across boundary sections is taken into account. The first domain solved is a B domain, which corresponds to the inlet section, the first membrane section, and the beginning of the second membrane section. Subsequent domains are C domains and correspond to

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

equations can be written for secondary variables (vorticity, , and stream function, ) and cartesian coordinates (x and z)

0.5 0.4

=

z

0.3 0.2

j 2  j2  + 2, jz2 jx

1 j j vx + vz = jx jz Re

0.1 0 0

25

50

75

100

vx =

x

0.05



 j2  j2  + 2 , jx 2 jz

j . jx

(10) (11) (12)

z

In this equation, x represents the normalized longitudinal direction, z the normalized transversal direction and Re the Reynolds number based on the mean inlet velocity and height of the cell: Re =

0 20

25

30 x

Fig. 5. (a) Grid for cells with four membrane sections. (b) Grid detail.

II I

III z lin=lf

Domain B xs IV

xf

ls

lf

V

IV

VII

lf

II Domain C V lf

IV

V

ls

lf

IV

VII

lf

Domain D V lf

V0 H , 

(13)

where  is the fluid density and  the fluid viscosity. This formulation of the Navier–Stokes equations is an interesting alternative when the flow is bi-dimensional and so it is commonly used in such cases. While the formulation in primary variables requires the solution of three equation, has no natural equation for the pressure and requires a complex iterative method, the formulation in secondary variables requires the solution of only two partial differential equations by a simpler iterative method. The dimensionless mass transport equation in cartesian coordinates is   2 jc jc j2 c 1 j c vx , (14) + + vz = jx jz P e jx 2 jz2 where Pe is the Peclet number

II VIII

(9)

j , jz

vz = −

VIII

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VI

VII

lout=ls+lf

Fig. 6. Schematic representation of domains B, C and D (for cells with more than four membrane sections) and respective boundaries.

the fully permeable sub-section of membrane section i − 1, the membrane section i and the beginning of membrane section i + 1. The last domain solved is a D domain and corresponds to the fully permeable sub-section of membrane section n − 1, the last membrane section and the outlet section. Each domain was discretized into a grid of nodes. A grid with 261 × 251 nodes distributed in a similar way to that represented in Fig. 5 was selected. 3.2. Equations and boundary conditions The flow in the cell is described by Navier–Stokes equations. For a bi-dimensional cell, the dimensionless Navier–Stokes

Pe =

V0 H . D

(15)

In this equation, D is the molecular diffusivity of the solute. The mass transport equation was solved by the so-called method (Miranda and Campos, 2001). In this method, a variable transformation =ln(c) is used to linearize the concentration gradients in the mass boundary layer over the membrane. The mass transport equation is converted into the transport equation     2  j j j 1 j2 j2 j 2 . (16) + 2 + vx + vz = 2 jx jz P e jx jz jx jz Conditions for boundaries I, II, III, VI and VII (Figs. 4 and 6) have been presented by Miranda and Campos (2000, 2001, 2002) and are listed in Table 2. The inflow values at boundary VIII of domains C and D are the outflow values of the previous sections (Table 3). Boundary conditions for the semi-permeable membranes (boundaries IV) were analysed by Miranda and Campos (2000,

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J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

is zero. The non-dimensional number v becomes

Table 2 Conditions for boundaries I, II, III, VI and VII I

II 3 2 2 [1 − 4(0.5 − z) ]

vx =

vz = 0 (0.5−z)3 2

= −  = 12(0.5 − z) 3(0.5−z) 2

jvx jz

III = 0 vx = 0

=0

VII

vx = 0

jvx jx = 0 jvz jx = 0 j jx = 0 j jx = 0 jc jx = 0 j jx = 0

vz = 0

vz = 0

vz = 0

=0 =0

 = 11 16  = jjvzx

 = 11 16 + −vz dx  = jjvzx

jc jz = 0 j jz = 0

c=1

jc jz = 0 j jz = 0

fv =

VI

jc jz = 0 j jz = 0



2001, 2002), but they will be briefly discussed in conjunction with the boundary conditions for the fully permeable membranes (boundaries V). The slip velocity at the interface between a flowing fluid and a porous material can be neglected for all pressure driven processes (Belfort and Nagata, 1985). Then, as long as membranes with pore sizes and porosities much larger than the ones of microfiltration membranes are not used as fully permeable membranes, the no-slip condition can be adopted for all membranes in the cell. The permeate velocity is given by v −vz (x, 0) = [1 − 0 c(x, 0)], 1 −  0

(17)

where • 0 represents the ratio between the osmotic pressure difference across a non-polarized membrane, 0 , and the static pressure difference across the membrane, Pm  0 =

0 . Pm

(18)

• v represents the ratio between the permeate velocity through a non-polarized membrane and the mean feed velocity v =

Pm − 0 , R m V0

(19)

where Rm is the membrane resistance. For more details about these non-dimensional numbers see Miranda and Campos (2001). For the fully permeable sub-sections the osmotic pressure difference across the membrane is zero and, consequently, 0 ,

f

Pm f

R m V0

,

(20)

where the superscript f indicates that the variables are for a fully permeable sub-section. Eq. (17) becomes −vz (x, 0) = fv .

(21)

On a semi-permeable membrane surface, the normal convective flux of mass promoted by suction is equal to the diffusive flux in the opposite direction promoted by the concentration gradient between the membrane surface and the bulk −

1 jc = −vz (x, 0)[c(x, 0) − cp ], P e jz

(22)

where cp is the normalized solute concentration in the permeate stream. For the fully permeable membrane, cp is equal to c(x, 0) and Eq. (22) simplifies −

1 jc = 0. P e jz

(23)

The boundary conditions for semi-permeable and fully permeable membranes are presented in Table 3. 3.3. Infinite number of membrane sections When n = ∞, the cell is equivalent to a cell with a single membrane which has two kinds of pores: pores which are fully permeable and pores which are permeable to the solvent but impermeable to the solute. Each kind of pore connects to an independent collector. The superficial velocity of the fluid that f crosses the fully permeable pores is equal to v and the superficial velocity of the fluid that crosses the semi-permeable pores is given by Eq. (19). The boundary conditions are those of Tables 2 and 3, except for boundaries IV and V. The velocity at the membrane surface is defined by −vz (x, 0) =

sv [1 − 0 c(x, 0)](1 − ) + fv , 1 −  0

(24)

where  is the fraction of the membrane which is fully permeable.

Table 3 Semi-permeable and fully permeable membrane boundary conditions IV vx = 0 −v z (x) =

f

v 1−0



[1 − 0 cm (x)]

 = 11 16 + −vz dx  = jjvzx − jjvxz jc − P1e jz = vm (cm − cp ) j − jz = 1

V

VIII

vx = 0

vxi (0, z) = vxi−1 (ls + lf , z)

−v z (x) = fv   = 11 16 + −vz  = jjvzx − jjvxz jc − P1e jz = 0 j jz = 0

vzi (0, z) = vzi−1 (ls + lf , z) dx

i (0, z) = i−1 (ls + lf , z) i (0, z) = i−1 (ls + lf , z) ci (0, z) = ci−1 (ls + lf , z)

i (0, z) = i−1 (ls + lf , z)

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

and for sufficiently large values of lin and lout :

Table 4 Boundary conditions for n = ∞

c = f2 (z, x, ls , lf , n, sv , s0 , Re, Sc, fv ).

IV–V vx = 0 −vz =

=

sv 1−0



(28)

According to Miranda and Campos (2002), the normal component of the velocity inside the mass boundary layer is usually equal to the opposite of the permeate velocity

[1 − 0 c](1 − ) + fv 

11 16 + −vz dx  = jjvzx − jjvxz jc −vz c = fv  c − P1e jz j f − jz = −vz − v 

−vz = vm .

(29)

Miranda and Campos (2002)showed that, if −vz = vm , Re and Sc can be substituted by their product, the Peclet number (Pe):

The mass of solute that crosses boundary IV–V is equal to the mass of solute that arrives at the membrane surface by convection minus the mass of solute that leaves the surface by diffusion into the bulk fv  c(x, 0) = −vz (x, 0) c(x, 0) +

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1 jc . P e jz

=

lf . ls + lf

(31)

And, since (26)

lm = n(ls + lf ).

Conditions for boundary IV–V are listed in Table 4.

lf and ls can be substituted in Eq. (30) by lm and 

3.4. Numerical solution of the equations

c = f4 (z, x, lm , , n, sv , s0 , P e, fv ).

The equations were solved by the finite-difference methods developed by Miranda and Campos (2001) with a small improvement: the convective terms of the equations were discretized by a hybrid scheme.

(32)

(33)

The concentration at the membrane surface is independent of the normal coordinate cm = f5 (x, lm , , n, sv , s0 , P e, fv ).

(34)

The velocity and concentration fields are function of the same non-dimensional numbers

4. Dimensional analysis The concentration field depends on geometrical and operational variables. The relevant geometrical variables are: the length of the inlet section, Lin , the length of a semi-permeable sub-section, Ls , the length of a fully permeable sub-section, Lf , the length of the outlet section, Lout , the number of membrane sections, n, and the height of the channel, H. The relevant operational variables are: the fluid viscosity, , the fluid dens , the fully sity, , the semi-permeable membrane resistance, Rm f permeable membrane resistance, Rm , the pressure difference across the semi-permeable membrane, Pms , the pressure diff ference across the fully permeable membrane, Pm , the mean feed velocity, V0 , the feed concentration, C0 , and the osmotic pressure in the feed stream, 0 . For simplicity, fluid density and viscosity were considered independent of the solute concentration. A dimensional analysis, based on the procedure developed by Miranda and Campos (2000), was done for the cell in study. If the pressure drop along the x direction is neglected, the concentration field can be expressed by the following functional relation: c = f1 (z, x, lm , lout , ls , lf , n, sv , s0 , Re, Sc, fv )

(30)

The length of the semi-permeable sub-sections and the length of the fully permeable sub-sections can be related through the fraction of the membrane which is fully permeable, , by the following equation:

(25)

Combining Eqs. (24) and (25) gives sv 1 jc [1 − 0 c(x, 0)](1 − )c(x, 0) = − . 1 −  0 P e jz

c = f3 (z, x, ls , lf , n, sv , s0 , P e, fv ).

(27)

v = g1 (x, z, lm , , n, sv , s0 , P e, fv ).

(35)

The permeate velocity and the surface concentration are also dependents of the same non-dimensional numbers vm = g2 (x, lm , , n, sv , s0 , P e, fv ).

(36)

For s0 = 0, the permeate velocity becomes independent of the concentration field and equal to sv over a semi-permeable f membrane and equal to v over a fully permeable membrane. For conventional cells (n = 1), the permeate velocity can be determined by taking the stagnant film equation (total retention is assumed): vm =

Shi ln(cm ), Pe

(37)

where Shi is the Sherwood number of the homologous impermeable cell. Theoretical analysis and numerical results (Miranda and Campos, 2002) show that the film equation is accurate for low and moderate values of sv and Pe. In conventional cells, the influence of sv , s0 and Peclet number on Sherwood number is well understood. Miranda and Campos (2002) performed this study for a conical cell and for

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Table 5 Parameters range Non-dimensional number

Range

Reference

fv sv 0

10−5 .10−2

10−4 10−4 0.1 100 4 106 0.1

10−5 .2 × 10−4 0.1; 0.5; 0.9 100 1 − 128 106 0.1

lm n Pe



a parallel plate cell. The results show that for a parallel plate cell, the values of the logarithmic Sherwood number defined by Shm ln =

vm P e ln(cm )

(38)

are identical to the values of the Sherwood number of the homologous impermeable cell with uniform mass production at the impermeable wall. The logarithmic Sherwood number increases with increasing x, Pe, sv and decreases with increasing s0 . 5. Results 5.1. Flow and concentration fields Flow and concentration numerical results were obtained for n=4 and for different values of the concentrate stream velocity, f v . All other variables were kept at reference values, most of which were taken from the literature, and they are representative of ultrafiltration and reverse osmosis processes (see Table 5 for the reference values used and Table 6 for two examples of hybrid membrane processes which have non-dimensional numbers belonging to the ranges studied). The streamlines in the vicinity of the membrane (z < 0.04) are represented in Fig. 7 f for three values of the concentrate stream velocity: v = f f 10−5 , v = 5 × 10−4 and v = 10−2 . Some of the streamlines link the cell inlet to the membrane surface, which means that part of the fluid flowing in the vicinity of the membrane leaves the cell by crossing either the semi-permeable membrane or the fully permeable membrane. The remaining streamlines link the inlet to the outlet of the cell, which means that the remaining fluid constitutes the retentate. f For the smallest value of v (Fig. 7a), the flow pattern is slightly disturbed by the flow that crosses the fully permeable membrane. As the velocity of the concentrate stream increases, the flow becomes increasingly disturbed (Figs. 7b and c). The analysis of the first and second membrane sections shows the streamlines abruptly changing direction just above the fully f permeable membrane, even for the largest v . The concentration field just above the membrane is represented in Fig. 8 by iso-concentration lines for the three values f f of v represented in Fig. 7. For low values of v , only the iso-concentration lines located near the membrane (with high

values) are disturbed by fluid recovery. Iso-concentration lines located far from the membrane (with low values) remain almost undisturbed. Some iso-concentration lines merge with the fully permeable membrane and then, at the beginning of the next semi-permeable membrane, slowly move away from the wall. This behavior reveals a small cleaning effect over the boundary layer. The cleaning effect increases as the velocity of the concentrate stream increases (Figs. 8b and c). Here, even iso-concentration lines located far from the membrane are f disturbed. For the highest values of v , the mass boundary layer is fully cleaned and starts growing afresh after each fully permeable membrane. This is illustrated by the re-emergence of the pattern of the iso-concentration lines from membrane section to membrane section (Fig. 8c). The boundary layer is cleaned by the fluid leaving the cell through the fully permeable membranes. This fluid follows the trajectories represented in Fig. 9. Most of it comes from a region of the cell where the solute concentration is equal to the feed concentration (c = 1). Near the membrane surface, this fluid crosses the mass boundary layer dragging solute by convection and, afterwards, permeates the fully permeable membrane. Since the concentrate stream cleans the mass boundary layer, the surface concentration decreases after each fully permeable membrane. Fig. 10 shows the surface concentration decreasing along each fully permeable membrane. The concentration decreases because the fluid that crosses the initial part of a fully permeable membrane comes from the most concentrated region of the mass boundary layer (see Fig. 10). Since the fluid recovery from the boundary layer lowers the surface concentration, the osmotic pressure over the surface decreases and, therefore, the permeate velocity increases. In Fig. 11, the permeate velocities in a conventional membrane (n = 1) and in a membrane with four sections (n = 4) are compared. The mean purity of the concentrate stream versus the velocity of the concentrate stream, for different values of 0 , is represented in Fig. 12. The mean purity of the concentrate decreases for increasing values of either 0 or the velocity of the concentrate stream. For high values of the velocity of the concentrate stream, the purity of the concentrate becomes independent of 0 and converges to 1, i.e., the concentration converges to the concentration of the inlet fluid. Two effects explain these results: • when 0 increases the concentration at the permeable surface decreases and, consequently, a smaller amount of solute is transported across the fully permeable membrane; • when the velocity of the concentrate stream increases, the solute from the boundary layer is diluted by a larger quantity of low concentration fluid. The influence of 0 and of the concentrate stream velocity in the mean solvent stream velocity is represented in Fig. 13. The mean velocity of the solvent stream increases slightly as the velocity of the concentrate stream increases, as explained before. This velocity sharply decreases as 0 increases, because the osmotic pressure converges to the static pressure

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

1223

Table 6 Non-dimensional numbers for a reverse osmosis/ultrafiltration hybrid cell and for a ultrafiltration/microfiltration hybrid cell

Semi-permeable membrane Fully permeable membrane Solute Solvent Solute concentration (C0 ) Viscosity () Density () Total membrane length (Lm ) Distance between parallel plates (H) s ) Resistance of the semi-permeable membrane (Rm f Resistance of the fully permeable membrane (Rm ) Pressuredrop across membranes Feed velocity Feed osmotic pressure Diffusivity

sv fv 0

Reynolds number Schmidt number Peclet number

Reverse osmosis/ultrafiltration hybrid cell

Ultrafiltration/microfiltration hybrid cell

Reverse osmosis membrane Ultrafiltration membrane NaCl Water 21900 ppm 1 × 10−3 kg m−1 s−1 1 × 103 kg m−3 0.1 m 0.001 m 1.34 × 1010 N s m−3 6.17 × 109 N s m−3 3.0 × 106 Pa 1.0 m s−1 1.5 × 106 Pa 1 × 10−9 m2 s−1 1.12 × 10−3 4.9 × 10−4 0.5 1000 1000 1 × 106

Ultrafiltration membrane Microfiltration membrane Dextran Water 0.1% w/w 1.0 × 10−3 kg m−1 s−1 1 × 103 kg m−3 0.1 m 0.001 m 6.17 × 109 N s m−3 6.0 × 108 N s m−3 1.2 × 105 Pa 0.1 m s−1 4.4 × 102 Pa 1.5 × 10−11 m2 s−1 1.9 × 10−4 2.0 × 10−3 3.6 × 10−3 98.2 6.79 × 104 6.67 × 106

difference across the semi-permeable membrane, i.e., the pressure driven gradient tends to zero. 5.2. Influence of sv Parameter sv represents the permeate velocity of a nonpolarized membrane and this can be increased by increasing the pressure difference across the membrane. Fig. 14 shows the iso-concentration lines for several values of sv and for the reference values of the other variables listed in Table 5. When sv increases, the concentration inside the mass boundary layer also increases. As this concentration increases, the surface concentration and the osmotic pressure also increase, and the velocity of the solvent stream decreases. Since the concentrate stream transports solute recovered from the mass boundary layer its purity must, therefore, increase. These results show that the purity of the concentrate stream f can be controlled by manipulating sv and v . In practice, sv f and v can be manipulated by adjusting the pressures in the f concentrate, solvent and retentate chambers. sv and v can be independently changed if the pressure in the solvent chamber is independent from the pressure in the concentrate chamber. Since the resistance to the flow across the fully permeable membrane is lower than the resistance across the semi-permeable membrane, the pressure in the concentrate collector must be above atmospheric pressure.

numerical results show that the surface concentration decreases for increasing velocity of the concentrate stream and increasing number of membrane sections. In a hybrid membrane cell with fully permeable membranes and high values of the f velocity of the concentrate stream (v = 10−2 ), the surface concentration along the length of the cell always remains below the maximum concentration in the first section. Fig. 15 shows the surface concentration in a conventional cell (n = 1), in a hybrid cell with four fully permeable sub-sections and in a hybrid cell with 32 fully permeable sub-sections. All of these cells have the same permeable length (lm = 90). The maximum dimensionless surface concentration is 5.75 for n = 1, 4.23 for n = 4 and 2.59 for n = 32. In the conventional cell (n = 1), 97% of the surface has a surface concentration above 2.59 and 82% above 4.23. 5.4. Influence of the number of membrane sections To study how the velocity of the solvent stream and the purity of the concentrate stream are influenced by the number of membrane sections, the separation process was simulated for several values of n. The fraction of fully permeable pores was kept constant and equal to 0.1 and, for each value of n, f several values of v were studied. Simulations were made for 0 = 0.1 and sv = 10−4 . The concentration along the surface of the membranes was used to calculate:

5.3. Control of the surface concentration Hybrid membranes can be useful for improving the selectivity of protein fractionation processes, because the polarization of the membrane can be significantly reduced. The

• the mean purity of the concentrate stream for each fully permeable sub-section, p¯ i ; • the mean concentration along each semi-permeable membrane, c¯mi .

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

0.04

0.04

0.03

0.03

1.06 1.12

0.02

0.02

1.27 1.54

z

z

1224

0.01

0.01

0

0 0

25

50

75

100

0

1.87 2.35 2.89 3.42

2.35 2.89 3.42 4.26

25

50

0.04

0.04

0.03

0.03

1.06 1.12

0.02

0.02

1.27

0.01

0.01

0

0 0

25

50

75

100

0

1.54 1.87 2.35 2.89 3.42

1.54 1.87 2.35 2.89 3.42

25

50

x

75

100

1.54 1.87 2.35 2.89 3.42

1.54 1.87 2.35 2.89 3.42

75

100

x

0.04

0.04

0.03

0.03

1.06 1.12

0.02

1.27

1.27

1.27

1.27

1.54 1.87 2.35 2.89 3.42

1.54

0.01

1.54 1.87 2.35 2.89 3.42

1.54 1.87 2.35 2.89 3.42

25

50

0.02

z

z

3.42 4.26

x

z

z

x

2.89 3.42 4.26

0.01 0

0 0

25

50

75

100

0

x

1.06

1.06 1.12

1.12

1.87 2.35 2.89 3.42

75

1.06 1.12

100

x

Fig. 7. Streamlines in a cell with four membrane sections, for increasing f f f f values of v : (a) v =10−5 , (b) v =5×10−4 , (c) v =10−2 (0 =0.1; s −4 v = 10 ; lm = 100;  = 0.1; Re = 100).

Fig. 8. Iso-concentration lines in a cell with four membrane sections, for f f f f increasing values of v : (a) v = 10−5 , (b) v = 5 × 10−4 , (c) v = 10−2 (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

These concentrations were compared with the concentration f for n = ∞ along the membrane. The results for v = 10−5 f −3 are represented in Fig. 16, for v = 10 in Fig. 17 and for f v = 10−2 in Fig. 18. f f For the intermediate value of v (v = 10−3 , Fig. 17), both c¯mi and p¯ i change with n. When the number of sections is increased, c¯mi decreases and p¯ i increases, converging to the same value

Figs. 16–18 show that both p¯ i and c¯mi converge to the same value, p¯ i∞ , as n converges to infinite. As a consequence, p¯ and ∞. c¯m must converge to p¯ ∞ and v¯m must converge to v¯m No matter what the velocity of the concentrate stream is, the highest efficiency of the cell is obtained for n = ∞. Two efficiencies can be defined, the solute recovery efficiency, solute

solute =

∞ = p¯ i∞ . c¯m i

and the solvent recovery efficiency, solvent

(39) f v

f (v

For the lower value of = 10−5 , Fig. 16), the number of sections does not affect the concentration over the semipermeable membranes because the amount of solute recovered from the boundary layer is too small. The purity of the concentrate stream increases with the number of sections. f f For higher value of v (v = 10−2 , Fig. 18), the number of sections has a large effect on the concentration over the semipermeable membranes and a small effect on the purity of the concentrate stream.

p¯ ∞ p¯ m

solvent =

v¯m . ∞ v¯m

(40)

(41)

Both efficiencies were studied as n was increased for several f values of v , and the results are represented in Figs. 19 and 20. The solute recovery efficiency increases with increasing values f of n and v , while the solvent recovery efficiency increases with increasing values of n but decreases with increasing values f f of v ; they become independent of v and converge to 1

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

1225

0.03

z

1.06

0.02

1.12 1.27

0.01

1.54 1.87 2.35 2.89

0 15

20

25

30

35

x Iso-concentration lines

Streamlines

Fig. 9. Iso-concentration lines and streamlines along the first of four membrane f sections (0 = 0.1; sv = 10−4 ; v = 10−2 ; lm = 100;  = 0.1; P e = 106 ).

Fig. 12. Mean purity of the concentrate stream as a function of the velocity f of the concentrate stream (v ) for a cell with four membrane sections and s different values of 0 (v = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

Fig. 10. Membrane surface concentration along a cell with four membrane sections and comparison with a conventional cell (0 = 0.1; sv = 10−4 ; f

v = 5 × 10−4 ; lm = 100;  = 0.1; P e = 106 ).

Fig. 11. Velocity of the solvent stream along the cell with four membrane sections and comparison with a conventional cell (0 = 0.1; sv = 10−4 ; f

v = 5 × 10−4 ; lm = 100;  = 0.1; P e = 106 ).

as n converges to ∞. It is not possible to maximize both of these efficiencies at the same time by changing the velocity of the concentrate stream.

Fig. 13. Mean velocity of the solvent stream as a function of the velocity f of the concentrate stream (v ) for a cell with four membrane sections and s different values of 0 (v = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

For small values of n, efficiencies can increase significantly, but, when n increases from 64 to 128 the increase in efficiencies is negligible. This means that the number of sections necessary to improve efficiencies above those for n = 64 is very high. These results can be explained by the analysis presented in Section 5.1. Large fully permeable sections are inefficient because the end part of a section recovers low concentration fluid from the outer regions of the boundary layer. If the same area is redistributed into a larger number of sections, each section becomes more efficient. As the number of sections increases, this redistribution effect becomes less and less significant. f When v converges to infinite, since the concentration along the membrane sections is never higher than the concentration at the end of the first section, a chart of the maximum concentration along the membrane as a function of n can easily be made from the chart of the surface concentration as a function

1226

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

z

0.04 0.03

1.02

0.02

1.06

0.01

1.12

1.12

1.19 1.19

1.19 1.27

1.19

1.27

1.27

0 0

25

50 x

75

100

0.04 1.06

z

0.03

1.12 1.27

0.02 0.01

1.54 1.87 2.35 2.89 3.42

2.35 2.89 3.42 4.26

25

50

0 0

2.35 2.89 3.42 4.26

75

Fig. 15. Surface concentration along the membranes, for different number f of membrane sections (n) (s0 = 0.1; sv = 10−4 ; v = 10−2 ; lm = 100;

 = 0.1; P e = 106 ).

2.89 3.42 4.26

100

x 0.04 1.06 1.12

z

0.03

1.27 1.54 1.87 2.35 2.89

0.02 0.01

4.26 5.58

0 0

25

2.35 2.89

2.89

4.26 5.58

4.26 5.58

50

75

2.89 4.26 5.58

100

x Fig. 14. Iso-concentration lines in a cell with four membrane sections, for increasing values of sv : (a) sv = 10−5 , (b) sv = 10−4 , (c) sv = 2 × 10−4 f

(0 = 0.1; v = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

Fig. 16. Mean surface concentration at each semi-permeable sub-section and mean purity of the concentrate at each fully permeable sub-section along the membrane for cells with different number of membrane sections (n) and f v = 10−5 (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

of x for n = 1 by defining the following variable: n=

lm . x

(42)

The same variable transformation can be used to obtain a chart of the mean concentration as a function of n from the chart of mean concentration as a function of x. The chart of the mean concentration as a function of x was obtained from the chart of the concentration as a function of x by the following expression:  cm (x) dx s  c¯m (x) = . (43) dx In Fig. 21 the mean concentration over the cell wall is repref sented as a function of n for several values of v . This figure shows that the mean surface concentration converges to f a single line when v converges to infinite. Therefore, for a cell with n membrane sections the cleaning effect (depolarf ization) reaches a maximum (and stabilizes) for values of v

Fig. 17. Mean surface concentration at each semi-permeable sub-section and mean purity of the concentrate at each fully permeable sub-section along the membrane for cells with different number of membrane sections (n) and f v = 10−3 (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

Fig. 18. Mean surface concentration at each semi-permeable sub-section and mean purity of the concentrate at each fully permeable sub-section along the membrane for cells with different number of membrane sections (n) and f v = 10−2 (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

1227

Fig. 21. Mean surface concentration for cells with different number of memf brane sections (n) and several values of v (0 =0.1; sv =10−4 ; lm =100; 6  = 0.1; P e = 10 ).

6. Conclusions

Fig. 19. Mean purity of the concentrate (p) ¯ normalized by the mean purity of the concentrate for an hybrid cell with an infinite number of sections (p¯ ∞ ) as a function of the number of membrane sections (n) for several values of f v (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

A hybrid membrane cell composed of semi and fully permeable membrane sub-sections was studied by numerical methods. The influence of several dimensional numbers was studied and the flow and concentration fields were characterized. The efficiency and productivity of the hybrid membrane cell are higher than that of a conventional cell. In the hybrid membrane cell the permeate velocity increases and the purity of the concentrate also increases, the membrane polarization decreases and a concentrate stream with a certain level of purity always forms. The optimal number of sub-sections, n, is infinite, i.e., a membrane with two kinds of pores which are uniformly distributed. However, no matter what the value of n, there is always a significant improvement in the separation performances when a hybrid cell is used. The concentration at the membrane surface can be controlled bellow a critical value, which may be useful in protein fractionation. The purity of the concentrate stream can also be controlled. Unfortunately, simultaneous optimization is impossible. Notation cm s c¯m s c¯m,i

Fig. 20. Mean permeate velocity (v¯m ) normalized by the mean permeate ∞ ) as a velocity for an hybrid cell with an infinite number of sections (v¯m f

function of the number of membrane sections (n) for several values of v (0 = 0.1; sv = 10−4 ; lm = 100;  = 0.1; P e = 106 ).

around 10−2 . This value optimizes the depolarization effect, however, (as described before) the purity of the concentrate stream is very low.

cp C C0 Cm s C¯ m

normalized solute concentration at the membrane surface normalized mean solute concentration at the surface of the semi-permeable membranes normalized mean solute concentration at the surface of the semi-permeable membrane of subsection i normalized permeate solute concentration solute concentration solute concentration at the feed stream solute concentration at the membrane surface mean solute concentration at the surface of the semi-permeable membranes

1228 s C¯ m,i

Cp D H lf lin lm lout ls L Lf Lin Lm Lout Ls n ni f ni in ni out ni s p p¯ p¯ i f

Pm Pms 0 Rm f Rm s Rm s v¯m s v¯m,i

vx vz V0 Vm V¯ms s V¯m,i

Vx Vz x

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229

mean solute concentration at the surface of the semi-permeable membrane of sub-section i permeate solute concentration diffusivity distance between parallel plates normalized length of each fully permeable subsection normalized length of the inlet section normalized total length of all membrane sections normalized length of the outlet section normalized length of each semi-permeable subsection length of the cell length of each fully permeable sub-section length of the inlet section total length of all membrane sections length of the outlet section length of each semi-permeable sub-section number of membrane sections number of nodes along the longitudinal direction for each fully permeable sub-section number of nodes along the longitudinal direction for the inlet section number of nodes along the longitudinal direction for the outlet section number of nodes along the longitudinal direction for each semi-permeable sub-section purity of the concentrate stream mean purity of the concentrate stream mean purity of the concentrate leaving subsection i static pressure difference across a fully permeable membrane static pressure difference across a semipermeable membrane osmotic pressure difference across the semipermeable membrane membrane resistance resistance of the fully permeable membranes resistance of the semi-permeable membranes normalized mean solvent velocity at the surface of the semi-permeable membranes normalized mean solvent velocity at the surface of the semi-permeable membrane of sub-section i longitudinal component of the velocity normalized normal component of the velocity mean feed velocity permeate velocity mean solvent velocity at the surface of the semipermeable sub-sections mean solvent velocity at the surface of the semipermeable membrane of sub-section i transversal component of the velocity normalized longitudinal component of the velocity normalized longitudinal coordinate

X Xf Xs z Z W

longitudinal coordinate local longitudinal coordinate in a fully permeable sub-section local longitudinal coordinate in a semipermeable sub-section normalized transversal coordinate transversal coordinate width of the cell

Non-dimensional numbers Sc Pe 0 s0 v f

Schmidt number Peclet number 0 Pm 0 Pms P Rm V0 f

v

Pm f Rm V0

sv Re Shi Shm ln

Pms −0 s V Rm 0

Reynolds number Sherwood number of an impermeable cell Logarithmic Sherwoodnumber

Greek letters 

solute

solvent 0    

fraction of the membrane which is fully permeable solute recovery efficiency solvent recovery efficiency feed osmotic pressure viscosity density ln(c) stream function vorticity

Acknowledgments The authors acknowledge the financial support given by FCT, POCTI/EQU/59724/2004. This work was also supported, via CEFT, by POCTI (FEDER). References Binning, R.C., 1961. Separation of mixtures. US Patent 2,981,680. Cheang, B., Zydney, A.L., 2004. A two-stage ultrafiltration process for fractionation of whey protein isolate. Journal of Membrane Science 231, 159–167. Belfort, G., Nagata, N., 1985. Fluid mechanics and cross-flow filtration: Some thoughts. Desalination 53, 57–79. Chung, K.Y., Belfort, G., Edelstein, W.A., Li, X., 1993. Dean vortices in curved tube flow: 5. 3-d MRI and numerical analysis in the velocity field. A.I.Ch.E. Journal 39, 1592–1602. Ghosh, R., 2003. Novel cascade ultrafiltration configuration for continuous, high-resolution protein–protein fractionation: a simulation study. Journal of Membrane Science 226, 85–99. Hinds, B.J., Chopra, N., Rantell, T., Andrews, R., Gavalas, V., Bachas, L.G., 2004. Aligned multiwalled carbon nanotube membranes. Science 303, 62–65.

J.M. Miranda, J.B.L.M. Campos / Chemical Engineering Science 62 (2007) 1215 – 1229 Kennedy, T.J., Merson, R.L., McCoy, B.J., 1974. Improving permeation flux by pulsed reverse osmosis. Chemical Engineering Science 29, 1927–1931. Kuberkar, V.T., Davis, R.H., 2001. Microfiltration of protein-cell mixtures with crossflushing or backflushing. Journal of Membrane Science 183, 1–14. Lee, H., Lightfoot, E.N., 1974. Boundary-layer skimming—calculation curiosity or promising process? A.I.Ch.E. Journal 20, 335–339. Levesley, J., Hoare, M., 1999. The effect of high frequency backflushing on the microfiltration of yeast homogenate suspensions for the recovery of soluble proteins. Journal of Membrane Science 158, 29–39. Liu, Y.-H.S., Livesay, E.A., Luebke, K.J., Johnston, S.A., 2003. A simple method to make a membrane with regions of differential permeability. Journal of Membrane Science 215, 95–101. Miranda, J.M., Campos, J.B.L.M., 2000. Concentration polarisation in a membrane placed under an impinging jet confined by a conical wall—a numerical approach. Journal of Membrane Science 182, 257–270. Miranda, J.M., Campos, J.B.L.M., 2001. An improved scheme to study mass transfer over a separation membrane. Journal of Membrane Science 188, 49–59. Miranda, J.M., Campos, J.B.L.M., 2002. Mass transfer in the vicinity of a separation membrane—the applicability of the stagnant film theory. Journal of Membrane Science 202, 137–150. Mitrovic, M., Radovanovic, F., 1984. Dual-membrane separation ii: Concentration polarization reduction in dual cells. The Chemical Engineering Journal 28, 59–63.

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Nitsche, L.C., Zhuge, S., 1995. Hydrodynamics and selectivity of antipolarization dialysis. Chemical Engineering Science 50, 2731–2746. Poyen, S., Quemeneur, F., Baroiu, B., Desille, M., 1987. Improvement of the flux of permeate in ultrafiltration by turbulence promoters. International Chemical Engineering 27, 441–447. Ray, R.R.W.W., Newbold, D., McCray, S., Friesen, D., Brose, D., 1991. Synergistic, membrane-based hybrid separation systems. Journal of Membrane Science 62, 347–369. Schubert, J.P., Todd, D.K., 1980. Hyperfiltration scoop apparatus and method. US Patent 4,218,314. Shaw, R.A., Deluca, R., Gill, W.N., 1972. Reverse osmosis: increased productivity by reduction of concentration polarization in laminar flow reverse osmosis using intermediate non-rejecting membrane sections. Desalination 11, 189–205. van Reis, R., Gadam, S., Frautschy, L.N., Orlando, S., Goodrich, E.M., Saksena, S., Kuriyel, R., Simpson, C.M., Pearl, S., Zydney, A.L., 1997. High-performance tangential flow filtration using charged membranes. Biotechnology and Bioengineering 56, 71–82. van Rijn, C.J.M., 2002. Nano and Micro Engineered Membrane Technology. Membrane Science and Technology Series 10.